Like everyone else I have to really think about muting because it's really kind of automatic. To better understand what I do when muting I have tried to list every muting technique I use and why, along with any warnings I can think of for each.
Technique-Place one or both hands over all the strings
When used-Between songs or when bass rests for several bars
Warnings-Don't have to worry about harmonics because you are covering a wide area of the strings.
Technique-Right hand fingers come to rest on next lower string while playing.
When used-All the time while playing fingerstyle.
Warnings-Only works for one string at a time.
Technique-Right thumb on string below where you are playing except when playing lowest string.
When used-At all times when playing
Warnings-Usually don't have to worry about harmonics except near the neck. Doesn't work for E (B on a fiver) string.
4)For Open String Play
Technique-3 above combined with using left hand fingers to mute the strings where the right thumb can't
When Used-Playing open strings
Warnings-Keep left hand near the nut away from the harmonics especially around 5 and 7 frets.
5)Pad of left fingers
Technique-You can mute the string above where you're fretting a note with the pad of your finger by rolling your finger slightly downward. Example, Playing a C with middle finger and muting the D string with the pad of middle finger. C continues to ring D is muted.
When used-As needed
Warnings-Watch for harmonics. Useful technique since with a very small movement you can mute and free the string quickly.
Technique-Lift freting finger slightly from the fret
When used-As needed
Warnings-Watch harmonics. Useful for bouncy rythym.
7)Optional right hand technique
Technique-Use right ring and pinky to mute higher strings above where you are plucking
When used-When crossing strings from high to low
Warnings-I find this one difficult and don't use it much
8)Mini rest/dead notes
Technique-This is more of a rythym technique but just prior to plucking a note I rest my right finger on the string momentarily to mute it prior to the next note.
When used-When rythym contains rests or dead notes
Warnings-Be careful of harmonics near the neck
Technique-don't worry about muting too much
When used-Whenever noise is desired
Warnings-Sometimes noise is Ok in a tune so you don't need to worry about muting or whether it's causing harmonics.
This is by no means all the possible muting techniques but it does cover most of the methods I use. I hope it's useful.